On estimating Schatten norm and power distances between quantum states

Abstract

We study the computational complexity of estimating the quantum Schatten $α$-norm distance ${\rm T}_α(ρ_0,ρ_1)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $ρ_0$ and $ρ_1$. This quantity serves as a lower bound on the trace distance and, for $α> 1$, is interchangeable with its powered version $Λ_α(ρ_0,ρ_1)$. For any constant $α> 1$, we develop an efficient rank-independent quantum estimator for ${\rm T}_α(ρ_0,ρ_1)$ with time complexity ${\rm poly}(n)$, achieving an exponential speedup over the prior best results of $\exp(n)$ due to Wang, Guan, Liu, Zhang, and Ying (TIT 2024). When $0<α<1$ is a constant, the quantum Schatten $α$-power distance $Λ_α(ρ_0,ρ_1)$ becomes a distance metric. Accordingly, we provide a rank-efficient quantum estimator for this quantity. Our quantum algorithm reveals a dichotomy in the computational complexity of the Quantum State Distinguishability Problem with Schatten $α$-norm (QSD $_α$), which involves deciding whether ${\rm T}_α(ρ_0,ρ_1)$ is at least $2/5$ or at most $1/5$. This dichotomy arises between the cases of $α> 1$ and $0 < α\leq 1$: 1. For any constant $α>1$, QSD$_α$ is $\sf BQP$-complete. 2. For any $1 \leq α(n) \leq 1+{\rm negl}(n)$, QSD$_α$ is $\sf QSZK$-complete, implying that no efficient quantum estimator for ${\rm T}_α(ρ_0,ρ_1)$ exists unless ${\sf BQP}={\sf QSZK}$. This $\sf QSZK$-hardness result also extends to the promise problem defined by $Λ_α(ρ_0,ρ_1)$ for constant $0<α<1$. The hardness results follow from reductions based on new rank-dependent inequalities for ${\rm T}_α(ρ_0,ρ_1)$ when $1\leq α\leq \infty$ and for $Λ_α(ρ_0,ρ_1)$ when $0<α<1$, which are of independent interest.